Modern Analysis (Studies in Advanced Mathematics)

Modern Analysis provides coverage of real and abstract analysis, offering a sensible introduction to functional analysis as well as a thorough discussion of measure theory, Lebesgue integration, and related topics. This significant study clearly and distinctively presents the teaching and research literature of graduate analysis:

Providing a fundamental, modern approach to measure theory

Investigating advanced material on the Bochner integral, geometric theory, and major theorems in Fourier Analysis Rn, including the theory of singular integrals and Milhin's theorem - material that does not appear in textbooks

Offering exceptionally concise and cardinal versions of all the main theorems about characteristic functions

Containing an original examination of sufficient statistics, based on the general theory of Radon measures
With an ambitious scope, this resource unifies various topics into one volume succinctly and completely. The contents span basic measure theory in an abstract and concrete form, material on classic linear functional analysis, probability, and some major results used in the theory of partial differential equations. Two different proofs of the central limit theorem are examined as well as a straightforward approach to conditional probability and expectation.
Modern Analysis provides ample and well-constructed exercises and examples. Introductory topology is included to help the reader understand such items as the Riesz theorem, detailing its proofs and statements. This work will help readers apply measure theory to probability theory, guiding them to understand the theorems rather than merely follow directions.

A lucid and effective presentation of...sophisticated material
- Joseph A. Cima, Department of Mathematics, University of North Carolina, Chapel Hill
Ambitious...Conversant...Great
Steven G. Krantz, Department of Math, Washington University, St. Louis, Missouri
The author has chosen his topics well to demonstrate how even the oldest subjects can have a modern approach that improves on the original. The text is all business and very readable, especially for the mathematically prepared reader. There is a lot to recommend from the use of this book, not the least of which is the fact that the reader participates in the continuing documentation of modern mathematical analysis.
-Timothy Hall, PQI Consulting